Exercise 30 Page 497

As we can see, △ ACD and △ ABE have two pairs of congruent angles. Now, recall the Angle-Angle (AA) Similarity Postulate.

Thanks to this, we know that △ ACD ~ △ ABE. Because of that, their corresponding side lengths are proportional. AC/AB = AD/AE Let's rewrite the numerators by using the Segment Addition Postulate. AC = AB + BC AD = AE + ED Next, let's substitute the equations above into the proportion equation.

We have obtained the required proportion.

Paragraph Proof

Given: & BE∥ CD Prove: & ABBC = AEED Proof: By the Corresponding Angles Theorem we get that ∠ ABE ≅ ∠ ACD and ∠ AEB ≅ ∠ ADC. Thus, △ ACD~△ ABE thanks to the AA Similarity Postulate. Then, we get the proportion ACAB = ADAE. The Segment Addition Postulates gives us AC = AB + BC and AD = AE + ED.

Substituting the last two equations into the proportion and performing the required operations, we get that ABBC = AEED, which is what we wanted.